Methods of Series Analysis. I. Comparison of Current Methods Used in the Theory of Critical Phenomena

Abstract

We survey the principal types of methods of series analysis which have been used in the study of critical phenomena with a view to determining their accuracy and applicability to the treatment of the critical-point singularities. These methods include the ratio method and its variants, such as the Neville-table method; the Padé-approximant procedures; and the procedures based on the generalized-Polya theorems of Thompson et al. We show that the actual procedures of Thompson et al. are mathematically equivalent to certain of the Padé-approximant procedures. We give a general error analysis for the series-analysis procedures and derive a relation between the expected magnitude of the errors in the parameters of $A{\mathrm{}(1-yx)\mathrm{}}^{-\gamma }$, namely, $\Delta y:\Delta \gamma :\Delta A$ as $1:J:J \ln J$, where $J$ is the order of the last term of the series analyzed. This relation is briefly illustrated by numerical data. We further give procedures for establishing estimates of the magnitude of errors in the parameters that have the same type of validity as those commonly used to determine the accuracy of a truncated Taylor series. We discuss the commonly occurring but anomalous case of "defects" (errant close pole and zero) in Padé-approximant procedures. We show them to be related to Padé's block structure of the approximant table and emphasize the artificial nature of the apparently rapid convergence that they cause. By numerical investigation of many test functions, similar in structure to those believed to appear in problems of critical phenomena, we have illustrated the following conclusions. First, for series where there is only a simple algebraic singularity, closer to the origin and well separated from any other singularity, the ratio method, perhaps with Neville-table improvement, is the most effective procedure. Second, for series where there are interfering singularities close to the one considered, or where there are singularities either closer than or nearly at the same distance from the origin as the one considered, the Padé procedures are best. Finally, for not exactly representable singularity structures of the type just decscribed, the convergence of even the Padé-approximant procedures are significantly slowed. None of the general methods described does a very impressive job in computing the $\gamma$ value if the function is in fact of the form $A{\mathrm{}(1-yx)\mathrm{}}^{-\gamma }\ln \mathrm{}|1-yx|\mathrm{}$.